SFWRTECH $4$MA$3$

Challenge Project $1$

A $nn$ Hilbert matrix has entries $H_{ij}=\frac{1}{i+j-1},$ so it has the form

$\left[\begin{array}{cccc}1& \frac{1}{2}& \frac{1}{3}& \text{c}\text{d}\text{o}\text{t}\text{s}\\ \frac{1}{2}& \frac{1}{3}& \frac{1}{4}& \text{c}\text{d}\text{o}\text{t}\text{s}\\ \frac{1}{3}& \frac{1}{4}& \frac{1}{5}& \text{c}\text{d}\text{o}\text{t}\text{s}\\ \text{:}& \text{:}& \text{:}& \text{d}\text{d}\text{o}\text{t}\text{s}\end{array}\right]$

For $n=2,3,$dots. $,$ generate a Hilbert matrix $H$ and a n$-$vector $b=Hx$ using $x,$ a n$-$vector with all

entries equal to $1.$ You can generate these via a function generatorHb. Using the

GaussElimination, forwardSubstitution, and backwardSubstitution functions developed in

Assignment $2,$ solve the system the system Hwidehat$(x)=b$ to obtain the approximate solution widehat$(x).$ Find

the $-$norm of the residual $r=b-$Hwidehat$(x)$ and of the error $x=$widehat$(x)-x,$ where $x$ is the true

solution, i$.$e$.,$ the n$-$vector with all entries equal to $1.$ How large can you take $n$ before the error is

$100\%?$ Also monitor the cond $(H)$ and see how this varies as a function of $n.$

Note:

Your PDf submission should include the complete algorithm package including the

following functions: GaussElimination, forwardSubstitution, backwardSubstitution,

generatorHb that work for a general $nn$ matrix and $n-$vector.

You cannot use a built$-$in Gauss elimination or LU decomposition functions of the

program.

For each $n,$ your program should print the following vectors: widehat$(x),x,r$

As part of your solution, you should present the following table in your PDF file

Finally, you should also identify the $n$ at which the error goes to $100\%$

To pass this challenge project, you must:

The data in point $3$ and the table in point $4$ in the Note above should be included in your

before your code in the pdf file.

Include the complete code for all the functions. Do not print images of the code or paste

code with a black background.

Obtain the correct answers for widehat$(x),x,r$ for the different values of $n$